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Séminaire Poincaré
The Genesis of the Theory of Relativity
Olivier Darrigol
CNRS : Rehseis
83, rue Broca
75013 Paris
The most famous of Albert Einstein’s papers of 1905 is undoubtedly the one concerning
the theory of relativity. Any modern physicist knows that this theory imposes a strict and general
constraint on the laws of nature. Any curious layman wonders at the daring reform of our ancestral
concepts of space and time. As often happens for great conceptual breakthroughs, the theory of
relativity gave rise to founding myths whose charm the historian must resist.
The first of this myth is that Einstein discovered the theory of relativity in a single stroke
of genius that defies any rational analysis. Some of Einstein’s reminiscences favor this thesis, for
instance his allusion to a conversation with Michele Besso in which he would have suddenly realized
that a reform of the concept of time solved long standing paradoxes of electrodynamics. One could
also argue that the historical explanation of a deep innovation is by definition impossible, since a
radically new idea cannot be derived from received ideas. In the case of Einstein’s relativity the
rarity of pre-1905 sources further discourages historical reconstruction, and invites us to leave this
momentous discovery in its shroud of mystery.
This romantic attitude does not appeal to teachers of physics. In order to convey some sort
of logical necessity to relativity theory, they have constructed another myth following which a
few experiments drove the conceptual revolution. In this empiricist view, the failure of etherdrift experiments led to the relativity principle; and the Michelson-Morley experiment led to the
constancy of the velocity of light; Einstein only had to combine these two principles to derive
relativity theory.
As a counterpoise to this myth, there is a third, idealist account in which Einstein is supposed
to have reached his theory by a philosophical criticism of fundamental concepts in the spirit of
David Hume and Ernst Mach, without even knowing about the Michelson-Morley experiment, and
without worrying much about the technicalities of contemporary physics in general.
A conscientious historian cannot trust such myths, even though they may contain a grain of
truth. He must reach his conclusions by reestablishing the contexts in which Einstein conducted
his reflections, by taking into account his education and formation, by introducing the several
actors who shared his interests, by identifying the difficulties they encountered and the steps they
took to solve them. In this process, he must avoid the speculative filling of gaps in documentary
sources. Instead of rigidifying any ill-founded interpretation, he should offer an open spectrum of
interpretive possibilities. As I hope to show in this paper, this sober method allows a fair intelligence
of the origins of relativity.
A first indication of the primary context of the early theory of relativity is found in the very
title of Einstein’s founding paper: “On the electrodynamics of moving bodies.” This title choice
may seem bizarre to the modern reader, who defines relativity theory as a theory of space and
time. In conformity with the latter view, the first section of Einstein’s paper deals with a new
kinematics meant to apply to any kind of physical phenomenon. Much of the paper nonetheless
deals with the application of this kinematics to the electrodynamics and optics of moving bodies.
Clearly, Einstein wanted to solve difficulties he had encountered in this domain of physics. A survey
of physics literature in the years 1895-1905 shows that the electrodynamics of moving bodies then
was a widely discussed topic. Little before the publication of Einstein’s paper, several studies with
similar titles appeared in German journals. Much experimental and theoretical work was being
done in this context. The greatest physicists of the time were involved. They found contradictions
between theory and experience or within theory, offered mutually incompatible solutions, and
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sometimes diagnosed a serious crisis in this domain of physics.
Since Heinrich Hertz’s experiments of 1887-8 on the electric production of electromagnetic
waves, Maxwell’s field theory was the natural frame for discussing both the electrodynamics and
the optics of moving bodies. In order to understand the evolution of this subject, one must first
realize that the theory that Maxwell offered in his treatise of 1873 widely differed from what is
now meant by “Maxwell’s theory.”
1 Maxwell’s theory as it was
Like most of his contemporaries, Maxwell regarded the existence of the ether as a fundamental and
undeniable fact of physics. He held this medium responsible for the propagation of electromagnetic
actions, which included optical phenomena in his view. His theory was a phenomenological theory
concerned with the macroscopic states of a continuous medium, the ether, which could combine
with matter and share its velocity v. These states were defined by four vectors E, D, H, B that
obeyed a few general partial differential equations as well as some relations depending on the
intrinsic properties of the medium. In the most complete and concise form later given by Oliver
Heaviside and Heinrich Hertz, the fundamental equations read
∇ × E = −DB/Dt ,
∇ ·D = ρ ,∇· B = 0 ,
∇ × H = j + DD/Dt
(1)
where j is the conduction current and D/Dt is the convective derivative defined by
D/Dt = ∂/∂ − ∇ × (v× ) + v(∇· ).
(2)
In a linear medium, the “forces” E and H were related to the “polarizations” D and B by the
relations D = E and B = µH, and the energy density (1/2)(E 2 + µH 2 ) of the medium had the
form of an elastic energy. For Maxwell and his followers, the charge density and the conduction
current j were not primitive concepts: the former corresponded to the longitudinal gradient of the
polarization or “displacement” D, and the latter to the dissipative relaxation of this polarization
in a conducting medium. The variation DD/Dt of the displacement constituted another form of
current. Following Michael Faraday, Maxwell and his disciples regarded the electric fluids of earlier
theories as a naı̈vely substantialist notion1
The appearance of the convective derivative D/Dt in Maxwell’s theory derives from his understanding of the polarizations D and B as states of a single medium made of ether and matter
and moving with a well-defined velocity v (that may vary from place to place): the time derivatives
in the fundamental equations must be taken along the trajectory of a given particle of the moving
medium. The resulting law of electromagnetic induction,
∇ × E = −DB/Dt = −∂B/∂t + ∇ × (v × B) ,
(3)
contains the (v × B) contribution to the electric field in moving matter. By integration around a
circuit and through the Kelvin-Stokes theorem, it leads to the expression
I
ZZ
d
E · dl = −
B · dS
(4)
dt
of Faraday’s law of induction, wherein the integration surface moves together with the bordering
circuit. When the magnetic field is caused by a magnet, the magnetic flux only depends on the
relative position of the magnet and the circuit so that the induced current only depends on their
relative motion.
In sum, the conceptual basis of Maxwell’s original theory widely differed from what today’s
physicists would expect. Electricity and magnetism were field-derived concept, whereas modern
1 J.C.
Maxwell, A treatise on electricity and magnetism, 2 vols. (Oxford, 1973); H. Hertz, “ Über die Grundgleichungen der Elektrodynamik für bewegte Körper,” Annalen der Physik, 41 (1890), 369-399.
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electromagnetism treats them as separate entities. A quasi-material ether was assumed. The fundamental equations (1) only correspond to our “Maxwell equations” in the case of bodies at rest,
for which the velocity v is zero and the convective derivative D/Dt reduces to the partial derivative ∂/∂t. One thing has not changed, however: the theory’s ability to unify electromagnetism and
optics. In a homogenous insulator at rest, Maxwell’s equations imply the existence of transverse
√
waves propagating at the velocity c = 1/ µ. Having found this electromagnetic constant to be
very close to the velocity of light, Maxwell identified these waves with light waves. The resulting
theory automatically excludes the longitudinal vibrations that haunted the earlier, elastic-solid
theories of optics.
Within a few years after Maxwell’s death (in 1879), a growing number of British physicists
saluted this achievement and came to regard Maxwell’s theory as philosophically and practically
superior to earlier theories. The Germans had their own theories of electricity and magnetism, based
on electric and magnetic fluids (or Amperean currents) directly acting at a distance. They mostly
ignored Maxwell’s theory until in 1888 Heinrich Hertz demonstrated the emission of electromagnetic
waves by a high-frequency electric oscillator. After this spectacular discovery was confirmed, a
growing number of physicists adopted Maxwell’s theory in a more or less modified form.
Yet this theory was not without difficulties. Maxwell had himself noted that his phenomenological approach led to wrong predictions when applied to optical dispersion, to magneto-optics,
and to the optics of moving bodies. In these cases he suspected that the molecular structure of
matter had to be taken into account.
2 Flashback: The optics of moving bodies
Maxwell’s idea of a single medium made of ether and matter implied that the ether was fully
dragged by moving matter, even for dilute matter. Whereas this conception worked very well
when applied to moving circuits and magnets, it was problematic in the realm of optics. The first
difficulty concerned the aberration of stars, discovered by the British astronomer James Bradley
in 1728: the direction of observation of a fixed star appears to vary periodically in the course of a
year, by an amount of the same order as the ratio (10−4 ) of the orbital velocity of the earth to the
velocity of light.2
The old corpuscular theory of light simply explained this effect by the fact that the apparent
velocity of a light particle is the vector sum of its true velocity and the velocity of the earth (see
Fig. 1). In the early nineteenth century, the founders of the wave theory of light Thomas Young and
Augustin Fresnel saved this explanation by assuming that the ether was completely undisturbed
by the motion of the earth through it. Indeed, rectilinear propagation at constant velocity is all
that is needed for the proof.3
Fresnel’s assumption implied an ether wind of the order of 30km/s on the earth’s surface,
from which a minute modification of the laws of optical refraction ought to follow. As Fresnel knew,
an earlier experiment of his friend François Arago had shown that refraction by a prism was in
fact unaffected by the earth’s annual motion. Whether or not Arago had reached the necessary
precision of 10−4 , Fresnel took this result seriously and accounted for it by means of a partial
dragging of the ether within matter. His theory can be explained as follows.
According to an extension of Fermat’s principle, the trajectory that light takes to travel
between two fixed points (with respect to the earth) is that for which the traveling time is a
minimum, whether the medium of propagation is at rest or not. The velocity of light with respect
to the ether in a substance of optical index n is c/n, if c denotes the velocity of light. The absolute
velocity of the ether across this substance is αu, where α is the dragging coefficient and u is the
absolute velocity of the substance (the absolute velocity being that with respect to the remote,
undisturbed parts of the ether). Therefore, the velocity of light along the element dl of an arbitrary
2 J. Bradley, “A new apparent motion discovered in the fixed stars; its cause assigned; the velocity and equable
motion of light deduced,” Royal Society of London, Proceedings, 35 (1728), 308-321.
3 A. Fresnel, “Lettre d’Augustin Fresnel à François Arago sur l’influence du mouvement terrestre dans quelques
phénomènes d’optique,” Annales de chimie et de physique, 9(1818), also in Oeuvres complètes, Paris (1868), vol. 2,
627-636.
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Figure 1: Stellar aberration. Suppose that the position of a fixed star in the sky is judged by the
orientation of a narrow straight tube through which it can be seen. If the earth is moving with
respect to the fixed stars at the velocity u, the latter sweeps the distance uτ during the time τ that
the light from the star takes to travel from the beginning to the end of the tube. Therefore, the
true light path makes a small angle with the direction of the tube. When the velocity of the earth is
perpendicular to the tube, this angle is θ ≈ tanθ = u/c. Owing to the annual motion of the earth,
the apparent position of the star varies with a period of one year.
trajectory is c/n + (α − 1)u · dl/ds with respect to the substance (with ds = kdlk). To first order
in u/c, the time taken by light during this elementary travel is
dt = (n/c)ds + (n2 /c2 )(1 − α)u · dl .
(5)
Note that the index n and the dragging coefficient in general vary along the path, whereas the
velocity u has the same value (the velocity of the earth) for the whole optical setting. The choice
α = 1 (complete drag) leaves the time dt and the trajectory of minimum time invariant, as should
obviously be the case. Fresnel’s choice,
α = 1 − 1/n2
(6)
dt = (n/c)ds + (1/c2 )u · dl ,
(7)
yields
so that the time taken by light to travel between two fixed points of the optical setting differs only
by a constant from the time it would take if the earth were not moving. Therefore, the laws of
refraction are unaffected (to first order) under Fresnel’s assumption.4
In 1846, the Cambridge professor George Gabriel Stokes criticized Fresnel’s theory for making
the fantastic assumption that the huge mass of the earth was completely transparent to the ether
wind. In Stokes’ view, the ether was a jelly-like substance that behaved as an incompressible fluid
under the slow motion of immersed bodies but had rigidity under the very fast vibrations implied
in the propagation of light. In particular, he identified the motion of the ether around the earth
with that of a perfect liquid. From Lagrange, he knew that the flow induced by a moving solid
(starting from rest) in a perfect liquid is such that a potential exists for the velocity field. From his
4 Cf. E. Mascart, Traité d’optique, 3 vols. (Paris, 1893), vol. 3, chap. 15. Fresnel justified the value 1 − 1/n 2 of
the dragging coefficient by making the density of the ether inversely proportional to the square of the propagation
velocity c/n (as should be in an elastic solid of constant elasticity) and requiring the flux of the ether to be conserved.
As Mascart noted in the 1870s, this justification fails when double refraction and dispersion are taken into account.
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Figure 2: Fizeau’s experiment. After reflection on a semi-reflecting blade, the light from the source
S is divided into two beams. The upper beam travels against the water stream in A 0 B0 , crosses the
lens L0 , is reflected on the mirror M, crosses L0 again, travels against the water stream in AB, and
returns to the semi-reflecting blade. The lower beam does the symmetrical trip, which runs twice
along the water stream. The phase difference between the two beams is judged from the interference
pattern in O.
recent derivation of the Navier-Stokes equation, he also knew that this property was equivalent to
the absence of instantaneous rotation of the fluid elements. Consequently, the propagation of light
remains rectilinear in the flowing ether, and the apparent position of stars in the sky is that given
by the usual theory of aberration.5
In order to account for the absence of effects of the earth’s motion on terrestrial optics,
Stokes further assumed that the ether adhered to the earth and had a negligible relative velocity
at reasonable distances from the ground.
To sum up, before the middle of the century, there were two competing theories of the optics
of moving bodies that both accounted for stellar aberration and for the absence of effects of the
earth’s motion on terrestrial optics. Fresnel’s theory assumed the stationary character of the ether
everywhere except in moving refractive media, in which a partial drag occurred. Stokes’ theory
assumed complete ether drag around the earth and irrotational flow at higher distances from the
earth.
In 1850 Hippolyte Fizeau performed an experiment in which he split a light beam into two
beams, had them travel through water moving in opposite directions, and measured their phase
difference by interference (see fig. 2). The result confirmed the partial drag of light waves predicted by Fresnel. Maxwell knew about Fizeau’s result, and, for a while, wrongly believed that it
implied an alteration of the laws of refraction by the earth’s motion through the ether. In 1864, he
performed an experiment to test this modification. The negative result confirmed Arago’s earlier
finding with improved precision. As Stokes explained to Maxwell, this result pleaded for, rather
than contradicted the Fresnel drag. Yet Maxwell remained skeptical about the validity of Fizeau’s
experiment. In 1867 he wrote:
This experiment seems rather to verify Fresnel’s theory of the ether; but the whole question of the state of the
luminiferous medium near the earth, and of its connexion with gross matter, is very far as yet from being settled
by experiment.
In this situation, it was too early to worry about an incompatibility between the electromagnetic
theory of light and the optics of moving bodies. In 1878, one year before his death, Maxwell still
judged Stokes’ theory “very probable.”6
5 G.G. Stokes, “On the aberration of light,” Philosophical magazine, 27(1845), 9-55; “On Fresnel’s theory of the
aberration of light,” ibid., 28 (1846), 76-81; “On the constitution of the luminiferous ether, viewed with reference
to the phenomenon of the aberration of light,” ibid., 29 (1846), 6-10. This result can be obtained from Fermat’s
principle, by noting that to first order the time taken by light to travel along the element of length dl has the form
dt = (1/c)ds − (1/c2 )v · dl (v denoting the velocity of the ether), so that its integral differs only by a constant (the
difference of the velocity potentials at the end points) from the value it would have in a stationary ether.
6 H. Fizeau, “Sur les hypothèses relatives à l’éther lumineux, et sur une expérience qui paraı̂t démontrer que
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Figure 3: The Michelson-Morley experiment. The light from the source S is divided into two beams
by the semi-reflecting blade R. After reflection on the mirrors M1 and M2 , the two beams return
to R. Their interference pattern is observed through the telescope T.
In the 1870s a multitude of experiments confirmed the absence of effect of the earth’s motion
on terrestrial optics. In 1874, the author of the best of those, Eleuthère Mascart, concluded:
The translational motion of the earth has no appreciable influence on optical phenomena produced by a terrestrial
source, or light from the sun, so these phenomena do not provide us with a means of determining the absolute
motion of a body, and relative motions are the only ones that we are able to determine.
Mascart and other continental experts interpreted this finding by means of Fresnel’s theory. British
physicists mostly disagreed, as can be judged from a British Association report of 1885 in which
a disciple of Maxwell criticized “Fresnel’s somewhat violent assumptions on the relation between
the ether within and without a transparent body.”7
In 1881 the great American experimenter Albert Michelson conceived a way to decide between
Fresnel’s and Stokes’ competing theories. Through an interferometer of his own, he compared the
time that light took to travel the same length in orthogonal directions (see fig. 3). If the ether was
stationary, he reasoned, the duration of a round trip of the light in the arm parallel to the earth’s
motion was increased by a factor [l/(c − u) + l/(c + u)]/(2l/c), which is equal to 1/(1 − u 2 /c2 ).
The corresponding fringe shift was about twice what his interferometer could detect. From the null
result, Michelson concluded that Fresnel’s theory had to be abandoned.8
A French professor at the Ecole Polytechnique,
Alfred Potier, told Michelson that he had
p
overlooked the increase of the light trip by 1/ 1 − u2 /c2 in the perpendicular arm of his interferometer. With this correction, the experiment became inconclusive. Following William Thomson’s
and Lord Rayleigh’s advice and with Edward Morley’s help, Michelson first decided to repeat
Fizeau’s experiment with his powerful interferometric technique. In 1886 he thus confirmed the
le mouvement des corps change la vitesse avec laquelle la lumière se propage dans leur intérieur,” Académie des
Sciences, Comptes-rendus, 33 (1851), 349-355; J.C. Maxwell, “On an experiment to determine whether the motion
of the earth influences the refraction of light,” unpub. MS, in Maxwell, The scientific letters and papers, ed. Peter
Harman, vol. 2 (Cambridge, 1995), 148-153; Maxwell to Huggins, 10 Jun 1867, ibid., 306-311; “Ether,” article for
the Encyclopedia Britannica (1878), reproduced ibid., 763-775.
7 E. Mascart, “Sur les modifications qu’éprouve la lumière par suite du mouvement de la source et du mouvement
de l’observateur,” Annales de l’Ecole Normale, 3 (1874), 363-420, on 420; R.T. Glazebrook, Report on “optical
theories,” British Association for the Advancement of Science, Report (1885), 157-261.
8 A. Michelson, “The relative motion of the earth and the luminiferous ether,” American journal of science, 22
(1881), 120-129.
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Fresnel dragging coefficient with greatly improved precision.9
At this critical stage, the Dutch theorist Hendrik Lorentz entered the discussion. He first
blasted Stokes’ theory by noting that the irrotational motion of an incompressible fluid around a
sphere necessarily involves a finite slip on its surface.10 The theory could still be saved by integrating
Fresnel’s partial drag, but only at the price of making it globally more complicated than Fresnel’s.
Lorentz therefore favored Fresnel’s theory, and called for a repetition of Michelson’s experiment of
1881after noting the error already spotted by Potier. Michelson and Morley fulfilled this wish in
1887 with an improved interferometer. The result was again negative, to every expert’s puzzlement:
while Fizeau’s experiment confirmed Fresnel’s theory, the new experiment contradicted it. 11
3 Lorentz’s theory
When in the early 1890s Hertz and Heaviside perfected Maxwell’s electrodynamics of moving
bodies, they noted that it was incompatible with Fresnel’s theory of aberration, but decided to
postpone further study of the relation between ether and matter. Unknown to them, Lorentz had
long ago reflected on this relation and reached conclusions that sharply departed from Maxwell’s
original ideas. Unlike Maxwell’s British disciples, Lorentz learned Maxwell’s theory in a reinterpretation by Hermann Helmholtz that accommodated the continental interpretation of charge,
current, and polarization in terms of the accumulation, flow, and displacement of electric particles.
In 1878 he gave a molecular theory of optical dispersion based on the idea of elastically bound
charged particles or “ions” that vibrated under the action of an incoming electromagnetic wave
and thus generated a secondary wave. For the sake of simplicity, he assumed that the ether around
the molecules and ions had exactly the same properties as the ether in a vacuum. He could thus
treat the interactions between ions and electromagnetic radiation through Maxwell’s equations in
a vacuum supplemented with the so-called Lorentz force.12
Using lower-case letters for the microscopic fields and Hertzian units, these equations read
∇ × e = −c−1 ∂b/∂t ,
∇ · e = ρm ,
f = ρm [e + c
∇ × b = c−1 [ρm v + ∂e/∂t] ,
∇·b=0 ,
−1
(8)
v × b] ,
where ρm denotes the microscopic charge density (confined to the ions) and f denotes the density
of the force acting on the ions. Note that there are only two independent fields e and b since
the constants and µ are set to their vacuum value. Although from a formal point of view these
equations can be seen as a particular case of the Maxwell-Hertz equations (1), they were unthinkable
to true Maxwellians who regarded the concepts of electric charge and polarization as emergent
macroscopic concepts and believed the molecular level to be directly ruled by the laws of mechanics.
Using his equations and averaging over a macroscopic volume element, Lorentz obtained
the first electromagnetic theory of dispersion. In 1892, he realized that he could perform similar
calculations when the dielectric globally moved through the ether at the velocity u of the earth.
He only had to assume that the ions and molecules moved through the ether without disturbing
it. Superposing the incoming wave and the secondary waves emitted by the moving ions, he found
that the resulting wave traveled at the velocity predicted by Fresnel’s theory. The partial ether drag
imagined by Fresnel was thus reduced to molecular interference in a perfectly stationary ether. 13
9 A. Michelson and E. Morley, “Influence of the motion of the medium on the velocity of light”, American journal
of science, 31 (1886), 377-386.
10 It seems dubious that Stokes, as an expert on potential theory in fluid mechanics, could have overlooked this
point. More likely, his jelly-like ether permitted temporary departures from irrotationality.
11 H.A. Lorentz, “De l’influence du mouvement de la terre sur les phénomènes lumineux,” Archives néerlandaises
(1887), also in Collected papers, 9 vols. (The hague, 1934-1936), vol. 4, 153-214; Michelson and Morley, “On the
relative motion of the earth and the luminiferous ether,” American journal of science, 34 (1887), 333-345.
12 Lorentz, “Over het verband tusschen de voortplantings sneldheit en samestelling der midden stofen,” Koninklijke Akademie van Wetenschappen, Verslagen (1878), transl. as “Concerning the relation between the velocity of
propagation of light and the density and composition of media” in Collected papers (ref. 11), vol. 2, 3-119.
13 Lorentz, “La théorie électromagnétique de Maxwell et son application aux corps mouvants,” Archives
néerlandaises (1892), also in Collected papers (ref. 11), vol. 2, 164-321.
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Notwithstanding with their global intricacy, Lorentz’s original calculations contained an interesting subterfuge. In order to solve equations that involved the wave operator
∂ 2 /∂x2 − c−2 (∂/∂t − u∂/∂x)2 in a reference frame bound to the transparent body, Lorentz introduced the auxiliary variables
x0 = γx ,
t0 = γ −1 t − γux/c2
that restored the form of the operator in the ether-bound frame for
p
γ = 1/ 1 − u2 /c2 .
(9)
(10)
He thus discovered the Lorentz transformation for coordinates (up to the Galilean transformation
x = x̄ − ut, where x̄ is the abscissa in the ether frame).14
A few months later, Lorentz similarly realized that to first order in u/c the field equations
in a reference frame bound to the earth could be brought back to the form they have in the ether
frame through the transformations
t0 = t − ux/c2 ,
e0 = e + c−1 u × b ,
b0 = b − c−1 u × e .
(11)
In other words, the combination of these transformations with the Galilean transformation x =
x̄ − ut leaves the Maxwell-Lorentz equations invariant to first order. Lorentz used this remarkable
property to ease his derivation of the Fizeau coefficient and to give a general proof that to first
order optical phenomena were unaffected by the earth’s motion through the ether.15
It is important to understand that for Lorentz the transformed coordinates and fields were
mathematical aids with no direct physical significance. They were only introduced to facilitate
the solution of complicated differential equations. The “local time” t0 was only called so because it depended on the abscissa. The true physical quantities were the absolute time t and
the fields e and b representing the states of the ether. In order to prove the first-order invariance of optical phenomena, Lorentz considered two systems of bodies of identical constitution, one
at rest in the ether, the other drifting at the velocity u. He first noted that to a field pattern
e0 = F (x, y, z, t), b0 = G(x, y, z, t) for the system at rest corresponded a field pattern e, b for the
drifting system such that e0 = F (x, y, z, t0 ), b0 = G(x, y, z, t0 ) (the abscissa x being measured in a
frame bound to the system). He then noted that e0 and b0 vanished simultaneously if and only if
e and b did so. Consequently, the borders of a ray of light or the dark fringes of an interference
pattern have the same locations in the system at rest and in the drifting system. The change of
the time variable is irrelevant, since the patterns observed in optical experiments are stationary.
We may conclude that Lorentz’s use of the Lorentz invariance was quite indirect and subtle.
There remained a last challenge for Lorentz: to account for the negative result of the
Michelson-Morley experiment of 1887. As George Francis FitzGerald had already done, Lorentz
noted that the frange shift expected in a stationary ether theory disappeared
if the longitudinal
p
arm of the interferometer underwent a contraction by the amount γ −1 = 1 − u2 /c2 when moving
through the ether. In order to justify this hypothesis, Lorentz first noted that in the case of electrostatics the field equations in a frame bound to the drifting body could be brought back to those
for a body at rest through the transformation x0 = γx. He further assumed that the equilibrium
length or a rigid rod was determined by the value of intermolecular forces and that these forces
all behaved like electrostatic forces when the rod drifted through the ether. Then the fictitious rod
obtained by applying the dilation x0 = γx to a longitudinally drifting rod must have the length that
this rod would have if it were at rest. Consequently, the moving rod contracts by the amount γ −1 .
The Lorentz contraction thus appears to result from a postulated similarity between molecular
forces of cohesion and electrostatic forces.16
14 Ibid.
: 297
“On the reflexion of light by moving bodies,” Koninklijke Akademie van Wetenschappen, Verslagen
(1892), also in Collected papers (ref.11), vol. 4, 215-218.
16 Lorentz, “De relative beweging van der aarde en den aether,” Koninklijke Akademie van Wetenschappen, Verslagen (1892), transl. as “The relative motion of the earth and the ether” in Collected papers (ref. 11), vol. 4,
220-223.
15 Lorentz,
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Fully explained in the Versuch of 1895, Lorentz’s theory gained broad recognition before
the end of the century. Two other physicists, Joseph Larmor of Cambridge and Emil Wiechert
of Königsberg, proposed similar theories in the same period. In the three theories, the basic idea
was to hybridize Maxwell’s theory with the corpuscular concept of electricity and to reduce every
optic and electromagnetic phenomenon to the interactions between electric particles through a stationary ether. Besides the optics of moving bodies, these theories explained a variety of magnetic
and magneto-optic phenomena, and of course retrieved the confirmed predictions of Maxwell’s
theory. They benefited from the contemporary rise of an experimental microphysics, including
the discoveries of x-rays (1895), radioactivity (1896), and the electron (1897). In 1896, the Dutch
experimenter Pieter Zeeman revealed the magnetic splitting of spectral lines, which Lorentz immediately explained through the precession of the orbiting charged particles responsible for the lines.
Being much lighter than hydrogen, these particles were soon identified to the corpuscle discovered
in cathode rays by Emil Wiechert and Joseph John Thomson. Following Larmor’s terminology,
this corpuscle became known as the electron and replaced the ions in Lorentz’s theory. 17
4 Poincaré’s criticism
In France, the mathematician Henri Poincaré had been teaching electrodynamics at the Sorbonne
for several years. After reviewing the theories of Maxwell, Helmholtz, Hertz, Larmor, and Lorentz,
he judged that the latter was the one that best accounted for the whole range of optic and electromagnetic phenomena. Yet he was not entirely satisfied with Lorentz’s theory, because he believed
it contradicted fundamental principles of physics. In general, Poincaré perceived an evolution of
physics from the search of ultimate mechanisms to a “physique des principes” in which a few general
principles served as guides in the formation of theories. Among these principles were three general
principles of mechanics: the principle of relativity, the principle of reaction, and the principle of
least action.18
For any believer in the mechanical nature of the electromagnetic ether, it was obvious that
these three principles applied to electrodynamics, since ether and matter were together regarded as
a complex mechanical system. In particular, it was clear that electromagnetic phenomena would be
the same if the same uniform boost was applied to the ether and all material objects. It the boost
was applied to matter only, effects of this boost were expected to occur. For instance, Maxwell
believed that the force between two electric charges moving together uniformly on parallel lines had
to vanish when their velocity reached the velocity of light. Poincaré thought differently. In his view,
the ether only was a convenient convention suggested by the analogy between the propagation of
sound and the propagation of light. In the foreword of his lectures of his lecture of 1887/8 on the
mathematical theories of light, he wrote:19
It matters little whether the ether really exists: that is the affair of the metaphysicians. The essential thing for us
is that everything happens as if it existed, and that this hypothesis is convenient for us for the explanation of the
phenomena. After all, have we any other reason to believe in the existence of material objects? That too, is only
a convenient hypothesis; only this will never cease to do so, whereas, no doubt, some day the ether will be thrown
aside as useless.
As we will see, Poincaré actually never abandoned the ether. But he refused to regard it as an
ordinary kind of matter whose motion could affect observed phenomena. In his view, the principle
of reaction and the principle of relativity had to apply to matter alone. In his lectures of 1899 on
Lorentz’s theory, he wrote: I consider it very probable that optical phenomena depend only on the
relative motion of the material bodies present –light sources and apparatus– and this not only to
first or second order but exactly.
17 Lorentz, Versuch einer Theorie der elektrischen un optischen Erscheinungen in bewegten Körpern (Leiden,
1895), also in Collected papers (ref. 11), vol.5, 1-139; “Optische verschinitjnelsen die met de lading en de massa der
ionen in verband staan,” Koninklijke Akademie van Wetenschappen, Verslagen (1898), transl. as “Optical phenomena
connected with the charge and mass of ions” in Collected papers (ref. 11) vol. 3, 17-39.
18 H. Poincaré, Electricité et optique. La Lumière et les théories électrodynamiques [Sorbonne lectures of 1888,
1890 and 1899)], ed. J. Blondin and E. Néculcéa, (Paris, 1901).
19 Poincaré, Théorie mathématique de la lumière (Sorbonne lectures, 1887-88), ed. J. Blondin (Paris, 1889), I.
10
O. Darrigol
Séminaire Poincaré
Figure 4: Cross-signaling between two observers moving at the velocity u through the ether. The
points A, A0 , A00 , B, B0 , B00 represent the successive positions of the observers in the ether when
the first observer sends a light signal, when the second observer receives this signal and sends back
another signal, and when the first observer receives the latter signal.
It must be emphasized that at that time no other physicist believed in this acceptance of the
relativity principle. Most physicists conceived the ether as a physical entity whose wind should have
physical effects, even though the precision needed to test this consequence was not yet available.
The few physicists, such as Paul Drude or Emil Cohn, who questioned the mechanical ether, felt
free to violate principles of mechanics, including the relativity principle.20
Lorentz’s theory satisfied Poincaré’s relativity principle only approximately and did so through
what Poincaré called two “coups de pouce”: the local time and the Lorentz contraction. Moreover,
it violated Poincaré’s reaction principle, since Lorentz’s equations implied that the net force acting
on all the ions or electrons should be the space integral of ∂(e × b)/c∂t, which does not vanish
in general. In his contribution to Lorentz’s jubilee of 1900, Poincaré iterated this criticism and
further discussed the nature and impact of the violation of the reaction principle. In the course
of this argument, about which more details will be given in a moment, he relied on Lorentz’s
transformations (11) to compute the energy of a pulse of electromagnetic radiation from the standpoint of a moving observer. The transformed fields e0 and b0 , he noted, are the fields measured
by a moving observer. Indeed the force acting on a test unit charge moving with the velocity u is
e + c−1 u × b = e0 according to the Lorentz force formula. Poincaré went on noting that the local
time t0 = t − ux/c2 was that measured by moving observers if they synchronized their clocks in
the following manner:21
I suppose that observers placed in different points set their watches by means of optical signals; that they try to
correct these signals by the transmission time, but that, ignoring their translational motion and thus believing that
the signals travel at the same speed in both directions, they content themselves with crossing the observations, by
sending one signal from A to B, then another from B to A.
Poincaré only made this remark en passant, gave no proof, and did as if it had already been
on Lorentz’s mind. The proof goes as follows. When B receives the signal from A, he sets his watch
to zero (for example), and immediately sends back a signal to A. When A receives the latter signal,
he notes the time τ that has elapsed since he sent his own signal, and sets his watch to the time
τ /2. By doing so he commits an error τ /2 − t− , where t− is the time that light really takes to
travel from B to A. This time, and that of the reciprocal travel are given by t − = AB/(c + u) and
t+ = AB/(c − u), since the velocity of light is c with respect to the ether (see fig. 4). The time
τ is the sum of these two traveling times. Therefore, to first order in u/c the error committed in
setting the watch A is τ /2 − t− = (t+ − t− )/2 = uAB/c2 . At a given instant of the true time, the
times indicated by the two clocks differ by uAB/c2 , in conformity with Lorentz’ expression of the
local time.
Poincaré transposed this synchronization procedure from an earlier discussion on the measurement of time, published in 1898. There he noted that the dating of astronomical events was
20 Poincaré,
ref. 18, 536.
“La théorie de Lorentz et le principe de la réaction.” In Recueil de travaux offerts par les auteurs à
H.A. Lorentz à l’occasion du 25ème anniversaire de son doctorat le 11 décembre 1900, Archives néerlandaises, 5
(1900), 252-278, on 272.
21 Poincaré,
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The Genesis of the Theory of Relativity
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based on the implicit postulate “that light has a constant velocity, and in particular that its velocity is the same in all directions.” He also explained the optical synchronization of clocks at rest,
and mentioned its similarity with the telegraphic synchronization that was then being developed
for the purpose of longitude measurement. As a member of the Bureau des Longitudes, Poincaré
naturally sought an interpretation of Lorentz’s local time in terms of cross-signaling. As a believer
in the relativity principle, he understood that moving observers would never know their motion
through the ether and therefore could only do as if these signals propagated isotropically. 22
Poincaré thus provided a physical interpretation of the transformed time t0 and the transformed fields e0 and b0 , which only referred to a fictitious system for Lorentz. This interpretation
greatly eased the use of this transformation, for it made the (first-order) invariance of optical
phenomena a direct consequence of the formal invariance of the Maxwell-Lorentz equations. Yet
it would be a mistake to believe that Poincaré thereby redefined the concepts of space and time.
In his terms, the Lorentz-transformed quantities referred to the apparent states of the field for a
moving observer. The true states remained those defined with respect to the ether. As we will see,
Poincaré never gave up this view.
5 The Lorentz invariance
Strangely, Lorentz overlooked Poincaré’s reinterpretation of his transformations, and kept reasoning
in terms of a fictitious system brought to rest. So did other experts on the electrodynamics of moving
bodies until at least 1904. Nevertheless, Lorentz took some of Poincaré’s criticism seriously. In 1904,
he offered a new version of his theory in which the invariance of optical phenomena held at every
order in u/c, without the ”coups de pouce” reproached by Poincaré. He knew since 1899 that the
homogenous field equations for a system bound to the earth could be brought to the form they
have for a system at rest in the ether through the transformations
x0 = γx, y 0 = y, z 0 = z, t0 = (γ −1 t − γuxc−2 )
e0 = −2 (1, γ)(e + c−1 u × b), b0 = −2 (1, γ)(b − c−1 u × e) ,
(12)
where is an undetermined constant (for a given value of u) and the factor (1, γ) means a multiplication by 1 of the component of the following vector parallel to u and a multiplication by γ
of the component perpendicular to u. In 1904, he generalized this result to the coupling between
electrons and field.23
Specifically, Lorentz realized that for a spherical electron subjected to the Lorentz contraction
and carrying the electromagnetic momentum
Z
p = c−1 (e × b)dτ ,
(13)
his transformations brought back the equation of motion
dp/dt = e[e + c−1 (u + v) × b]
(14)
of an electron of charge e to the form it has for a system at rest (u = 0), if an only if the constant had the value 1. On his way to this result, he derived the expression p = m0 γv of the momentum,
where m0 = e2 /6πRc2 is the electromagnetic mass of a spherical-shell electron of radius R. Lastly,
Lorentz gave expressions of the transformed source terms of the field equations such that dipolar
emission in the moving system transformed into dipolar emission in the system at rest. Combining,
all these results he could assert that optical phenomena in a moving system were the same as in a
system at rest.
This result only held in the dipolar approximation, because Lorentz’s expression of the transformed source terms was not the one today regarded to be correct. Lorentz also neglected the
22 Poincaré,
“La mesure du temps”, Revue de métaphysique et de morale, 6 (1898), 371-384.
phenomena in a system moving with any velocity smaller than light,” Royal Academy
of Amsterdam, Proceedings (1904), also in Collected papers (ref. 11), vol. 5, 172-197.
23 Lorentz,“Electromagnetic
12
O. Darrigol
Séminaire Poincaré
spinning motion of the electrons, and overlooked the cohesive forces that the stability of his contractile electron required. His derivation of the invariance of optical phenomena was complex and
indirect, for it involved a double-step transformation, the fictitious system at rest, and comparison
between the states of this system and those of the real system. For other phenomena, there is
no doubt that Lorentz still believed that motion with respect to the ether could in principle be
detected.
Poincaré reacted enthusiastically to Lorentz memoir, because he saw in it an opportunity to
satisfy the relativity principle in a complete and exact manner. He published the results of the
ensuing reflections under the title “Sur la dynamique de l’électron,” first as a short note of 5 June
1905 in the Comptes-Rendus, and as a bulky memoir in the Rendiconti of the Circolo matematico
di Palermo for the following year. He first defined the “relativity postulate” as follows:
It seems that the impossibility of experimentally detecting the absolute motion of the earth is a general law of
nature; we naturally incline to assume this law, which we shall call the Postulate of Relativity, and to do so without
any restriction.
Correcting Lorentz’s expression of the transformed source terms, he then showed that “the Lorentz
transformations”
x0 = γ(x − ut), y 0 = y, z 0 = z, t0 = γ(t − uxc−2 ) ,
e0 = (1, γ)(e + c−1 u × b, b0 = (1, γ)(b − c−1 u × e) ,
(15)
left the Maxwell-Lorentz equations invariant. These transformations are obtained by combining
the transformations (12), which Lorentz used, with the Galilean transformation x0 = x − ut.
Poincaré showed that they formed a group, and used this property to determine the global scaling
factor . He noted that the coordinate transformations left the quadratic form x 2 + y 2 + z 2 − c2 t2
invariant and could thus be regarded as rotations in a four-dimensional space with an imaginary
fourth coordinate. He obtained the relativistic law for the “addition” of velocities, for which the
combined velocity always remains inferior to the limit c.24
Next, Poincaré showed that a model of the contractile electron could be conceived in which the
cohesive forces (the so-called Poincaré tension) preserved the Lorentz invariance. He thus retrieved
Lorentz’s expression p = m0 γv for the momentum of the electron. Lastly, he argued that in order
to be compatible with the postulate of relativity, gravitational interactions should propagate at
the velocity of light; and he proposed modifications of Newton’s law of gravitation that made it
compatible with Lorentz invariance.
Thus, there is no doubt that Poincaré regarded Lorentz invariance as a general requirement
for the laws of physics, and that he identified this formal condition with the principle of relativity.
On the latter point, his only comment was:
The reason why we can, without modifying any apparent phenomenon, confer to the whole system a common
translation, is that the equations of an electromagnetic medium are not changed under certain transformations
which I shall call the Lorentz transformations; two systems, one at rest, the other in translation, thus become exact
images of one another.
The Palermo memoir, long and thorough as it was, said nothing on the interpretation to be given
to the transformed coordinates and fields. Perhaps Poincaré believed this should not be the main
point. Perhaps he had not yet been able to provide an operational understanding of Lorentz’s local
time at any order in u/c. There is no doubt, however, that he regarded the transformed fields and
coordinates as the ones measured by moving observers. At the Saint-Louis conference of 1904, he
repeated (and attributed to Lorentz!) his definition of the local time by optical cross-signaling. In
his Sorbonne lectures of 1906, he proved that this definition remained valid at any order in u/c,
and he characterized the Lorentz transformations as the ones giving the “apparent space and time
24 Poincaré, “Sur la dynamique de l’électron,” Académie des Sciences, Comptes-rendus, 140, (1905), 1504-1508;
“Sur la dynamique de l’électron,” Rendiconti del Circolo matematico di Palermo (1906), also in Poincaré, Oeuvres
(Paris, 1954), vol. 9, 494-550, on 495.
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The Genesis of the Theory of Relativity
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Figure 5: Poincaré’s light ellipsoid (a = OA, b = OB, f = OF ).
coordinates.”25
The same lectures and later talks on the “mécanique nouvelle” show that Poincaré nonetheless
maintained the ether and the ordinary concepts of space and time. In his view, the clocks bound
to the ether frame gave the true time, for it was only in this frame that the true velocity of light
was c. The clocks of a moving frame only gave the apparent time. For those who would think that
the difference with Einstein’s theory of relativity is merely verbal, it is instructive to look at an
argument Poincaré repeatedly gave to justify optical synchronization. 26
Simultaneity should be transitive, namely: if the clock A is synchronized with the clock B,
and if the clock B is synchronized with the clock C, then the clock A should be synchronized
with clock C for any given choice of the positions of the three clocks. Indeed, any breakdown of
transitivity could be used to detect motion through the ether and thus to violate the relativity
principle. Now consider an observer moving with the constant velocity u through the ether and
emitting a flash of light at time zero. At the value t of the true time, this light is located on a
sphere of radius ct centered at the emission point. Poincaré next considered the appearance of this
light shell for a moving observer, the rulers of which are subjected to the Lorentz contraction. The
result is an ellipsoid of revolution,
p the half-axes of which have the values a = γct and b = ct (see
fig. 5). As the eccentricity is e = 1 − b2 /a2 = u/c, the focal distance f = ea = γut is equal to the
apparent distance traveled by the observer during the time t. Therefore, the Lorentz contraction
is the contraction for which the position of the observer at time t coincides with the focus F of the
light ellipsoid he has emitted.
Now consider a second observer traveling with the same velocity u and receiving the flash
of light at the time t+ . The position M of this observer belongs to the ellipsoid t = t+ , and the
distance F M represents the apparent distance between the two observers, which is invariable.
According to a well-known property of ellipses, we have
F M + eF P = b2 /a ,
(16)
where P denotes the projection of M on the larger axis. The length F P being equal to the difference
x0 of the apparent abscissas of the two observers, this implies
t+ = γF M/c + γux0 /c2 .
(17)
Suppose that the two observers synchronize their clocks by cross-signaling. The traveling time of
the reverse signal is
t− = γF M/c − γux0 /c2
(18)
25 Poincaré, ibid., 495; “L’état actuel et l’avenir de la physique mathématique” (Saint-Louis lecture), Bulletin des
sciences mathématiques, 28 (1904), 302-324, transl. in Poincaré, The foundations of science (New York, 1929); “Les
limites de la loi de Newton,” Sorbonne lectures (1906-1907) ed. by H. Vergne in Bulletin astronomique publié par
l’observatoire de Paris, 17 (1953), 121-365, chap. 11.
26 Poincaré, ibid., 218-220; “La dynamique de l’électron,” Revue générale des sciences pures et appliquées (1908),
also in Oeuvres (ref. 24), vol. 9, 551-586.
14
O. Darrigol
Séminaire Poincaré
Therefore, two events are judged simultaneous by these observers if and only if their true times
differ by
(t+ − t− )/2 = γux0 /c2 .
(19)
This condition is obviously transitive.27
For any one familiar with Einstein’s theory of relativity, this reasoning seems very odd. Indeed,
the light ellipsoid corresponds to a fixed value of the time t in one reference frame and to space
measured in another frame. In general, Poincaré’s theory allows for the use of the “true time”
in any reference system, whereas our relativity theory regards this sort of mixed reference as a
mathematical fiction. This means that the conceptual basis of Poincaré’s theory is not compatible
with Einstein’s, even though both theories are internally consistent and have the same empirical
predictions (for the electrodynamics of moving bodies).28
Another oddity of Poincaré’s theory is his naming the Lorentz contraction ”a hypothesis.”
As we just saw, Poincaré showed that the contraction was necessary to the transitivity of optical
synchronization, which itself derives from the relativity principle. He nonetheless spoke of a hypothesis, probably because he did not quite trust the implicit conventions made in this reasoning.
In the Palermo memoir, he clearly indicated his dissatisfaction with the present state of the theory:
We cannot content ourselves with simply juxtaposed formulas that would agree only by some happy coincidence; the
formulas should, so to say, penetrate each other. Our mind will be satisfied only when we believe that we perceive
the reason of this agreement, so that we may fancy that we have predicted it.
Poincaré meant that the Lorentz covariance of all forces in nature, including gravitation, could
not be regarded as a mere consequence of the principle of relativity. He believed this symmetry
also implied more arbitrary assumptions, such as the similarity between electromagnetic and other
forces and the universality of the velocity of light as a propagation velocity. 29
As Poincaré reminded his reader, one way to justify these assumptions was the electromagnetic
view of nature, according to which electromagnetism should be the ultimate basis of all physics.
More appealing to him was the following suggestion:
The common part of all physical phenomena would only be an appearance, something that would pertain to our
methods of measurement. How do we perform our measurements? By superposing objects that are regarded as
rigid bodies, would be one first answer; but this is no longer true in the present theory, if one assumes the Lorentz
contraction. In this theory, two equal lengths are, by definition, two lengths which light takes equal time to travel
through. Perhaps it would be sufficient to renounce this definition so that Lorentz’s theory would be as completely
overturned [bouleversée] as Ptolemy’s system was through Copernicus’ intervention.
Lorentz’s explanation of the null result of the Michelson-Morley experiment, Poincaré reasoned,
implicitly rested on the convention that two lengths (sharing the same motion) are equal if and
only if light takes the same (true) time to travel through them. What he meant by dropping
this convention is not clear. Some commentators have speculated that he meant a revision of the
concept of time, in Einstein’s manner. This is not very likely, because the context of Poincaré’s
suggestion was length measurement instead of time measurement, and also because he ignored
Einstein’s point of view to the end of his life. More likely he was alluding to a suggestion he had
earlier made at the Saint-Louis conference: “that the ether is modified when it moves relative to
the medium which penetrates it.”30
To sum up, in 1905/6 Poincaré obtained a version of the theory of relativity based on the
principle of relativity and the Lorentz group. He believed this symmetry should apply to all forces
in nature. He exploited it to derive the dynamics of the electron on a specific model and to suggest
27 Although Poincaré did not do so much, the expression of the local time t 0 can simply be obtained by requiring
the apparent velocity of light to be equal to c. This condition implies F M = ct0 , and t = γ(t0 + ux0 /c2 ). Calling
x the true abscissa of the second observer at time t (with respect to the emission point of the flash), we also have
x0 = γ(x − ut). Consequently, t0 = γ(t − ux/c2 ), in conformity with the Lorentz transformations.
28 The empirical equivalence of the two theories simply results from the fact that any valid reasoning of Einstein’s
theory can be translated into a valid reasoning of Poincaré’s theory by arbitrarily calling the time, space, and fields
measured in one given frame the true ones, and calling all other determinations apparent.
29 Poincaré, ref. 24 (1906), 497.
30 Poincaré, ref. 24 (1906), 498; ref. 25 (1904), 315 (Foundations).
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The Genesis of the Theory of Relativity
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a modification of the law of gravitation. He nevertheless maintained the ether as the medium
in which light truly propagated at the constant velocity c and clocks indicated the true time.
He regarded the quantities measured in moving frames as only apparent, although the principle
of relativity forbade any observational distinction between a moving frame and the ether frame.
He understood the compatibility of the Lorentz transformations of coordinates with the optical
synchronization of clocks and the invariance of the apparent velocity of light, but hesitated on the
physical significance of the Lorentz contraction and never discussed the dilation of time.
6 Einstein’s theory
Albert Einstein had an early interest in electrodynamics, if only because his family owned a small
electrotechnical company. At age sixteen, he wrote a little essay on the state of the ether in an
electromagnetic field. If we believe a late reminiscence, he also wondered about the appearance of
a light wave for an observer traveling along with it. In 1896 he entered the Zürich Polytechnikum,
where he learned electrodynamics in the standard continental style. Two years later he studied
Maxwell’s theory by himself from Drude’s Physik des Aethers. Drude was a sympathizer of Ernst
Mach’s philosophy, and belonged to a tradition of German physics that favored phenomenological
theories over mechanistic assumptions. In his rendering of Maxwell’s theory, he avoided any picture
of ether processes and propounded to redefine the ether as space endowed with special physical
properties.31
In the summer of 1899, Einstein’s reading of Hertz’s Untersuchungen prompted the following
comment, addressed to his lover Mileva Marić:
I am more and more convinced that the electrodynamics of moving bodies, as it is presented today, does not agree
with the truth, and that it should be possible to present it in a simpler way. The introduction of the name ’ether’
into the electric theories has led to the notion of a medium of whose motion one could speak of without being able,
I believe, to associate a physical meaning to this statement. I believe that electric forces can be directly defined
only for empty space, [which is] also emphasized by Hertz. Further, electric currents will have to be regarded not as
’the vanishing of electric polarization in time’ but as motion of true electric masses, whose physical reality seems to
result from the electrochemical equivalents.... Electrodynamics would then be the science of the motions in empty
space of moving electricities and magnetisms.
Einstein’s criticism targeted Hertz’s electrodynamics of moving bodies, which developed the
Maxwellian idea of an ether fully dragged by matter. Einstein was also aware of the Maxwellian
concept of electric current as “the vanishing of electric polarization in time,” and suggested to replace it with the motion of ions moving through empty space. What he had in mind probably was
a theory similar to Lorentz’s and Wiechert’s, in which electromagnetic phenomena are brought
back to the interactions of ions through a stationary ether. Einstein added that “the radiation
experiments” would decide between the two conceptions. He presumably meant to compare the
intensities of light emitted from the same source in opposite directions.32
A month later, Einstein thought of another experiment concerning “the effect that the relative
motion of bodies with respect to the luminiferous ether has on the velocity of propagation of light
in transparent bodies.” The physics professor, Heinrich Weber, to whom he explained this project
and the motivating theory, told him to read a paper by Wilhelm Wien that contained a short
account of Lorentz’s theory and a description of many experiments on ether motion, including
those of Fizeau and of Michelson-Morley. Einstein presumably welcomed Fizeau’s result, which
confirmed the stationary ether. But he may have doubted the import of the Michelson-Morley
null result, because two years later, in the fall of 1901, he was still planning a new interferometric
method “for the search of the relative motion of matter with respect to the luminiferous ether.” 33
31 A. Einstein, “Über die Untersuchung des Aetherzustandes im magnetischen Felde,” in John Stachel et al.
(eds), The collected papers of Albert Einstein, vol. 1 (Princeton, 1987), 6-9; P. Drude, Physik des Aethers auf
electromagnetischer Grundlage (Stuttgart, 1894).
32 Einstein to Marić [Aug 1899], ECP 1, 225-227.
33 Einstein to Marić, 10 and 28? Sep 1899, ECP 1, 229-230, 233-235; W. Wien, “ Über die Fragen, welche die
translatorische Bewegung des Lichtethers betreffen,” appendix to Annalen der Physik, 65 (1898): I-XVIII); Einstein
to Grossmann, 6 Sep 1901, in Stachel (ref. 31), 315-316.
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O. Darrigol
Séminaire Poincaré
Sometime after 1901, Einstein ceased to look for experimental tests of motion through the
ether and adopted the relativity principle. Although the null-result of ether-drift experiments
probably contributed to this move, Einstein’s autobiographical remarks and the relativity paper
of 1905 give an essential role to another kind of consideration. In the introduction to this paper,
Einstein remarks that magneto-electric induction receives two very different interpretations in
Lorentz’s theory, according as it is the magnet or the electric conductor that is moving with
respect to the ether. In the first case the motion of the magnet implies the existence of an electric
field e within the conductor (such that ∇ × e = −∂b/∂t)). In the second case, there is no electric
field within the conductor, and the Lorentz force (v × b per unit charge) is responsible for the
motion of the electrons. Yet the induced current only depends on the relative motion of the coil
and the magnet.34
This fairly obvious remark had already been made by several authors. Most of these, however, easily accepted the theoretical asymmetry and believed that finer details of electromagnetic
induction or other phenomena would reveal effects of motion through the ether. Einstein thought
differently. Following an epistemological trend expressed in Hertz’s, Hume’s, and Mach’s writings,
he rejected theoretical asymmetries that had no empirical counterpart. As Lorentz’s stationary
ether led to many asymmetries of this sort, it had to be rejected.
This way or reasoning explains why Einstein, unlike Poincaré, conflated the adoption of the
relativity principle with the rejection of the ether. He thus found himself compelled to imagine a
theory of electromagnetic propagation that would respect the relativity principle without disturbing
the confirmed predictions of Lorentz’s theory. The relativity principle implies that the measured
velocity of light should be the same in any inertial frame. For one who has given up the ether,
there is a simple way to satisfy this requirement: to make this velocity depend on the velocity
of the emitter, as was the case in Newton’s old corpuscular theory of light. According to later
reminiscences, this was Einstein’s first idea: he tried to modify the expression of the retarded
interaction between two charged particles in such a way that it would depend on their relative
motion only. Alas, difficulties soon came up. In particular, Einstein found that the light emitted
by an accelerated source could sometime back up on itself, because the successive wave planes
traveled at different velocities depending on the velocity of the source during their emission. 35
Einstein gave up this theory, and long remained unable to conciliate Lorentz’s theory with the
relativity principle. According to the Kyoto lecture of 1922, he suddenly realized that a redefinition
of the concept of time solved his problem during a conversation with his friend Michele Besso in
the spring of 1905. In the same lecture, Einstein also indicated that he had earlier tried to assume
the validity of the Maxwell-Lorentz equations in any inertial frame. This assumption of course
implied that the velocity of light should be the same in any inertial frame, against the Galilean
rule for the addition of velocities. The difficulty disappeared when Einstein realized that there was
“an inseparable connection between time and signal velocity.”36
This remark and the ensuing developments may have been eased by Einstein’s readings. We
surely know he had read Lorentz’s Versuch of 1895, and was therefore aware of the local time and
the role it served in preserving the form of the Maxwell-Lorentz equations to first order. He may
also have known the exact form of the Lorentz transformations, for in 1904 several German theorists
commented on them in journals that he regularly read. We also know that he read Poincaré’s La
science et l’hypothèse, which contained an eloquent plead for the relativity principle as well as a
brief criticism of simultaneity:
There is no absolute time. To say two durations are equal is an assertion which has by itself no meaning and which
can acquire one only by convention. Not only have we no direct intuition of the equality of two durations, but we
have not even direct intuition of the simultaneity of two events occurring in different places: this I have explained
34 A. Einstein, “Zur Elektrodynamik bewegter Körper,” Annalen der Physik, 17 (1905), 891-921, on 891. Einstein
removed this asymmetry in 1905 (ibid. on 909-910) by making the separation between electric and magnetic field
depend on the reference frame and by defining the electromotive force in a conductor as the electric field in a frame
bound to this conductor.
35 Einstein to Ehrenfest, 25 Apr 1912, quoted in John Stachel et al. (eds.) The collected papers of Albert Einstein,
vol. 2 (Princeton, 1989), 263, and further ref. ibid.
36 Einstein, “How I created the theory of relativity?” (from notes taken by Jun Ishiwara from Einstein’s lecture
in Kyoto on 14 Dec 1922), Physics today, 35: 8 (1982), 45-47.
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17
in an article entitled La mesure du temps.
The German version of this book, published in 1904 and perhaps the one read by Einstein, had a
long citation of Poincaré’s article of 1898, including:
The simultaneity of two events or the order of their occurrence, and the equality of two time intervals must be
defined so that the expression of the laws of physics should be the simplest possible; in other words, all those rules
and definitions [conventions for time measurement] only are the fruits of an unconscious opportunism.
The German editor further mentioned the possibility that a new time coordinate may be a function
of the older time and space coordinates. Lastly, Einstein may have read Poincaré’s memoir of 1900,
which contained the interpretation of Lorentz’s local time in terms of optically synchronized clocks.
Or he may have been aware of Emil Cohn’s remark of 1904 that the local time was the time for
which the propagation of light was isotropic.37
Whether Einstein borrowed this idea or rediscovered it by himself, he became aware of a
relation between local time and optical synchronization. In his understanding of the relativity
principle, there was no ether and all inertial frames were entirely equivalent. Therefore, the time
and space coordinates defined in these frames all had the same status. The constancy of the
velocity of light no longer resulted from the existence of the ether, and had to be postulated
separately. According to the relativity principle, this property had to hold in any inertial system.
The apparent absurdity of this consequence disappeared if time was defined in conformity with
the light postulate. This definition turned out to imply the Lorentz transformations, without any
recourse to the Maxwell-Lorentz equations.
Through reasoning of that kind, Einstein arrived at the “new kinematics” that formed the first
part of his celebrated memoir of 1905 “On the electrodynamics of moving bodies.” The following
summary should suffice to exhibit the magnificent architecture of this memoir.38
Introduction: Einstein exposes the aforementioned asymmetry of electromagnetic induction in Lorentz’s theory and uses it to plead for the strict validity of the relativity
principle. He announces that this principle, together with the principle of the constancy
of the velocity of light, leads to a new kinematics that solves the contradictions of the
electrodynamics of moving body.
I. Kinematical part
§1. For two distant clocks A and B of identical constitution attached to a given reference frame,
Einstein defines synchronicity by the condition tB − tA = t0A − tB , where tA is the time of the
clock A at which a light signal is sent from A, tB the time of the clock B at which this signal
reaches B and a replying signal is sent from B, and t0A the time at which the latter signal
arrives at A. This definition is arranged so that the propagation of light should be isotropic
in the given frame. “In conformity with experience,” Einstein further assumes that the ratio
2AB/(t0A − tA ) is the universal constant c.
§2. Einstein states the two principles on which his new kinematics is built: the “relativity principle” according to which the laws of physics are the same in any inertial system, and the
“principle of the constancy of the velocity of light” in a given inertial system. He then shows
that simultaneity is a relative notion, because according to the above given criterion two
clocks synchronized in a given reference frame are not in another.
§3. Einstein derives the Lorentz transformations by requiring the velocity of light to be the same
constant in two different frames of reference in the two following cases: when the light path
is parallel to the relative velocity of the two frames, and when it is normal to this velocity in
one of the frames. His reasoning also relies on the group structure of the transformations in
order to determine the global scaling factor of the transformations.
37 Poincaré,
38 Einstein,
La science et l’hypothèse (Paris, 1902), 111; Wissenschaft und Hypothese (Leipzig, 1904), 286-289.
ref. 34.
18
O. Darrigol
Séminaire Poincaré
§4. Einstein gives the “physical consequences” of these transformations for the behavior of moving rigid bodies and clocks. The extremities of a rigid ruler moving edgewise at the velocity
v in a given frame of reference coincide,pat a given instant of this frame, with points of this
frame whose distance is proportional to 1 − v 2 /c2 . Einstein thus introduces the contraction
of lengths as a perspectival effect. Most innovatively, he shows that a clock C traveling at
the uniform speed v from the clock A to the clock
p B of a given reference frame appears to be
slow compared to these clocks by a factor 1/ 1 − v 2 /c2 . He predicts the same retardation
if the clock C makes a U-turn at B and returns to A. He extends this result to any loop-wise
trip of the clock C.
§5. Einstein gives the relativistic law for the composition of velocities.
II. Electrodynamic part
This part is devoted to the application of the new kinematics to electrodynamics.
§6. Einstein proves the covariance of the homogenous Maxwell-Lorentz equations, and uses the
relevant field transformations to remove the theoretical asymmetry he condemned in his
introduction: the force acting on a moving unit point charge at a given instant must now be
regarded as the electric field acting on it in an inertial frame that has the same velocity as
the charge does at this instant.
§7. Einstein uses the transformation of a plane monochromatic wave to derive the Doppler effect
and stellar aberration.
§8. Einstein derives the transformation law for the energy of a light pulse, and uses this law to
derive the work done by radiation pressure on a moving mirror.
§9. Einstein obtains the covariance of the inhomogeneous Maxwell-Lorentz equations, thus establishing “the conformity of the electrodynamic basis of Lorentz’s theory. . . with the relativity
principle.” This remark implies that the new theory retrieves every consequence of Lorentz’s
theory for the electrodynamics and optics of moving bodies (including the Fresnel drag, for
instance).
§10. Einstein obtains the relativistic equation of motion of an electron in an electromagnetic field
by assuming the approximate validity of Newtonian mechanics in a quasi-tangent frame (in
which the velocity of the electron remains small within a sufficiently small time interval)
and transforming to the laboratory frame. He calls for experimental testing of the resulting
velocity dependence of the mass of the electron. As he knew, the Göttingen experimenter
Walther Kaufmann had performed several experiments of that kind in order to test the
existence of an electromagnetic mass and to decide between competing models of the electron.
Most of the components of Einstein’s paper appeared in others’ anterior works on the electrodynamics of moving bodies. Poincaré and Alfred Bucherer had the relativity principle. Lorentz
and Larmor had most of the Lorentz transformations, Poincaré had them all. Cohn and Bucherer
rejected the ether. Poincaré, Cohn, and Abraham had a physical interpretation of Lorentz’s local
time. Larmor and Cohn alluded to the dilation of time. Lorentz and Poincaré had the relativistic
dynamics of the electron. None of these authors, however, dared to reform the concepts of space
and time. None of them imagined a new kinematics based on two postulates. None of them derived
the Lorentz transformations on this basis. None of them fully understood the physical implications
of these transformations. It all was Einstein’s unique feat.
7 The inertia of energy
In the fall of 1905, Einstein wrote to his friend Conrad Habicht:
Another consequence of the electrodynamics paper came to my mind. Together with Maxwell’s fundamental equations, the relativity principle implies that mass is a measure of the energy content of bodies. Light transports mass.
Vol. 1, 2005
The Genesis of the Theory of Relativity
19
There should be a sensible diminution of mass in the case of radium. The line of thought is amusing and fascinating.
But is not the dear Lord laughing about it? Is not he pulling me by the nose? This much I cannot know.
Although Einstein’s extraordinary conclusion was entirely new, the paradoxes that led to it were
not. In order to see that, we need to return to Poincaré’s memoir of 1900 on the reaction principle
in Lorentz’s theory.39
Remember that Poincaré denounced the violation of this principle when applied to matter
alone. Lorentz and other theorists were unshaken by this objection. They believed that the ether,
stationary though it was, could well carry the missing momentum. In the name of the electromagnetic worldview, Max Abraham based his dynamics of the electron of 1902 on the concept of
electromagnetic momentum. Ironically, he attributed the expression c−1 e × b of this momentum
to Poincaré, who had only introduced it as an absurd contribution to the momentum balance.
Poincaré’s abhorrence for this notion was not a mere consequence of his ghostly concept of
the ether. It resulted from his Newtonian insight that any violation of the principle of reaction led,
together with the relativity principle, to the possibility of perpetual motion. Suppose, with him
and Newton, that two bodies, initially at rest and isolated from other bodies, act on each other
in a non-balanced way by forces that depend only on their configuration. Connect the two bodies
by a rigid bar. The resulting system begins to move. According to the principle of relativity, the
net force acting on the system does not depend on the acquired velocity. Therefore, the system
undergoes a forever accelerated motion.
This reasoning, which assumes direct action from matter to matter, does not immediately
apply to electrodynamics. In this case, Poincaré examined the implications of Lorentz’s theory for
the motion of the center of mass of the matter-field system. Calling m the mass density of matter,
and j the energy density (e2 + b2 )/2 of the field, the Maxwell-Lorentz equations imply the global
momentum relation40
Z
Z
mvdτ + c−1 e × bdτ = constant ,
(20)
and the local energy relation
∂j
+ ∇ · (ce × b) = −ρv · e .
∂t
In turn, these relations imply the balance
Z
Z
Z
d
−2
c jrdτ + mvdτ + (c−2 ρv · e)rdτ = constant .
dt
(21)
(22)
When there is no energy transfer between matter and field, the third term vanishes and the theorem
of the uniform motion of the center of mass of the system is saved by associating to the field the
flow of a fictitious fluid of mass density j/c2 . In the general case, Poincaré added the fiction of a
latent, ether-bound fluid. He set the local conversion rate between free and latent fluid to ρv · e/c 2 ,
so that the center of mass of matter, free fluid, and latent fluid moved uniformly. It must be
emphasized that he only introduced these fictitious entities to show more precisely how Lorentz’s
theory violated the theorem of the center of mass.
Poincaré went on to show that this violation, or the concomitant violation of the reaction
principle, led to absurd consequences. For this purpose, he considered a Hertzian oscillator placed
at the focus of a parabolic mirror and emitting radiation at a constant rate. This system moves
with the absolute velocity u in the direction of emission, and is heavy enough so that the change of
this velocity can be neglected for a given momentum change. For an observer at rest in the ether,
the conservation of energy reads
S = J + (−J/c)u ,
(23)
where S is the energy spent by the oscillator in a unit time, J the energy of the emitted wave
train, and −J/c the recoil momentum according to Lorentz’s theory. For an observer moving at the
39 Einstein
to Habicht, undated [Jun to Sep 1905], in The collected papers of Albert Einstein, vol. 3; Poincaré, ref.
21.
40 In
1900, Poincaré had no reason to doubt the expression mv of the momentum density of matter. He assumed
that eventual non-electromagnetic forces balanced each other.
20
O. Darrigol
Séminaire Poincaré
velocity u of the emitter, the recoil force does not work, and the spent energy S is obviously the
same. According to the Lorentz transformations for time and fields (to first order), this observer
should ascribe the energy J(1 − u/c) to the emitted radiation, and the value (−J/c)(1 − u/c)
to the recoil momentum. Hence the energy principle is satisfied for the moving observer, but the
momentum law is not. Poincaré concluded that the violation of the principle of reaction in Lorentz’s
theory led to a first-order violation of the relativity principle for electromagnetic forces. 41
In his Saint-Louis lecture of 1904, Poincaré acknowledged recent experimental confirmations of
the radiation pressure, as well as Kaufmann’s measurements of the velocity-dependence of the mass
of the electron. The latter results, he now judged, “rather seemed to confirm. . . the consequences
of the theory contrary to Newton’s principle [of reaction].” In 1906, he explained how this violation
could be conciliated with the relativity principle at the electronic level: the lack of invariance of
the force acting on an electron under the Lorentz transformations is compensated by the velocitydependence of its mass, so that the equations of motion written in different reference frames are
equivalent. More generally, Poincaré argued that if all forces, including inertial ones, transformed
like electromagnetic forces under a Lorentz transformation, then the balance of forces held in any
reference frame.42
As everything worked fine at the microscopic level, Poincaré did not feel necessary to revisit
the macroscopic radiation paradox of 1900. Had he done so, he would have seen that the discrepancy
Ju/c2 between the recoil force in the moving frame and in the ether frame could not be explained
by any velocity dependence of the mass of the emitter, since the velocity change of the emitter
is then negligible. With the benefit of hindsight, we may note that a decrease of the mass of the
emitter by J/c2 during the emission process solves the paradox. Indeed, in the ether frame this
mass decrease implies a modification of the momentum variation of the emitter by (J/c 2 )u, which
exactly compensates the force Ju/c2 .
In his Annalen article of September 1905 (published in November), Einstein considered a
radiation process from the point of view of two different observers, as Poincaré had done in 1900.
The only difference is that he considered a symmetric process, which avoids any consideration of
recoil momentum. For an observer bound to the emitter, the same energy J/2 is emitted by the
light source in two opposite directions. For an observer moving at the velocity u with respect to the
source on the emission line, Einstein’s earlier transformation rule for the energy of light pulses gives
γ(1 + u/c)J/2 for the energy emitted in one direction and γ(1 − u/c)J/2 for the energy emitted
in the other. The sum of these energies exceeds the energy J by J(γ − 1). As the kinetic energy
of the emitter is the product of its mass by (γ − 1)c2 , a variation −J/c2 of this mass during the
emission restores the energy balance. From this remark, Einstein jumped to the general conclusion
that “the energy of a body depends on its energy content.”43
As appears from the letter to Habicht cited above, the inference was so novel that Einstein
was not yet quite sure about it. The following year, he argued that mass-energy equivalence was
the necessary and sufficient condition for an extension of the theorem of the center of mass to
electromagnetic systems. At the beginning of this memoir, he noted that Poincaré’s memoir of
1900 contained “the simple formal considerations on which the proof of this assertion is based.”
Indeed, Poincaré’s equation
Z
Z
Z
d
c−2 jrdτ + mvdτ + (c−2 ρv · e)rdτ = constant
(22)
dt
results from the equation
d
dt
Z
c
−2
jrdτ +
Z
mrdτ = constant
(24)
for the uniform motion of the center of mass if the mass density of matter follows the increase
41 In 1898, Alfred Liénard had already pointed to the first-order modification of the Lorentz force through a Lorentz
transformation.
42 Poincaré, ref. 25 (1904), 310 (Foundations); ref. 24 (1906), 490, 503.
43 Einstein, “Ist die Trägheit eines Körper von seinem Energieinhalt abhängig ?” Annalen der Physik, 18 (1905),
639-641.
Vol. 1, 2005
The Genesis of the Theory of Relativity
21
ρv · e of its energy content according to
Dm/Dt ≡ ∂m/∂t + ∇ · (mv) =< ρv · e/c2 > ,
(25)
wherein the symbols < > indicate an average at the scale at which the mass density m is defined.
As Einstein noted, the reasoning only makes sense if the expression of mv the momentum density
is applicable and if the only relevant energies are the energy of the electromagnetic field and the
internal energy of matter. This can be the case if the mass density m is defined at a macroscopic
scale for which each volume element includes many molecules and if the macroscopic kinetic energy
remains small.44
More concretely and more in the spirit of Poincaré’s perpetual motion argument, Einstein
considered an emitter and an absorber of radiation that faced each other and belonged to the same
solid. He imagined the following cycle:
• The source emits a radiation pulse with the energy J in the direction of the absorber, which
implies a recoil momentum for the solid.
• When the pulse reaches the absorber, the solid returns to rest.
• A massless carrier then brings the energy J back to the absorber.
At the end of this cycle, the solid has shifted by the amount −(J/M c)L/c (in a first approximation),
where M is the mass of the solid and L the distance between emitter and absorber. In order to
avoid the resulting sort of perpetual motion, Einstein assumed that the return of the energy J
to the emitter involved a transfer of mass J/c2 . During this transfer, the center of mass of the
global system does not move. Therefore, the solid moves by the amount L(J/c 2 )/M (in a first
approximation), which compensates the shift in the first step of the cycle.
This was neither the last nor the most convincing of Einstein’s derivation of the inertia of
energy. Einstein nonetheless believed in this astonishing consequence of relativity theory.
Conclusions
The genesis of the theory of relativity was a long process that involved at least three key players
and their critical reflections on the electrodynamics of moving bodies. Although Maxwell made
optics a part of electrodynamics, he could not explain optical phenomena that depend on the
molecular structure of matter. Toward the end of the nineteenth century, four circumstances favored investigations of this issue: Hertz’s confirmation of Maxwell’s theory; continental attempts
to inject into this theory the molecular conception of electricity that belonged to the defeated
German theories; the rise of an experimental microphysics of ions, x-rays, and electrons; and the
multiplication of experiments on the optics of moving bodies. A number of theorists then improved
the electrodynamics of moving bodies in competing approaches.
For the sake of simplicity, we may extract from this thriving physics the main events that
contributed to the formation of relativity theory. A first essential step was taken by Lorentz, who
replaced Maxwell’s hybrid, macroscopic, ether-matter medium with a stationary ether in which
electrons and other atomistic entities freely circulated. Exploiting the invariance properties of the
fundamental equations for the interaction between electrons and fields, Lorentz accounted for the
absence of effects of the motion of the earth through the ether, but only to a certain approximation.
Poincaré made this absence of effects a general postulate. He gave a physical interpretation of the
Lorentz transformations as those giving the space, time, and fields measured by moving observers.
He obtained the exact form of these transformations. He used them to determine electron dynamics
and to suggest a modification of Newton’s theory of gravitation. Yet he maintained the ether as
a privileged frame in which true time and space were defined. Einstein adopted the relativity
principle, eliminated the ether, and placed the space and time determinations in any two inertial
44 Einstein, “Das Prinzip der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie,” Annalen der
Physik, 20 (1906), 627-633.
The reasoning can easily be extended to fast moving material elements: one only has to
p
replace m with m/ 1 − v 2 /c2 .
22
O. Darrigol
Séminaire Poincaré
systems on exactly the same footing. Combining the relativity principle with that of the constancy
of the velocity of light, he obtained the Lorentz transformations, the contraction of lengths, and
the dilation of times. He showed how this symmetry permitted a consistent electrodynamics of
moving bodies and determined the dynamics of the electron. He derived the inertia of energy.
The construction of the special theory of relativity did not end with Einstein’s papers of 1905.
Some features that today’s physicists judge essential were added only later. For example, Hermann
Minkowski and Arnold Sommerfeld developed the 4-dimensional notation and the relativistic tensor formulation of electromagnetism; Max Planck gave the relativistic definition of force and the
Lagrangian formulation of relativistic dynamics; Max von Laue gave the kinematical interpretation
of the Fresnel drag as a direct consequence of the relativistic combination of the velocity of the
moving transparent body and the velocity of light within it.
Thus, Einstein was neither the first nor the last contributor to relativity theory. He learned
much by reading the best authors of his time, and he partly duplicated results already obtained by
Lorentz and Poincaré. Yet there is no doubt that his papers of 1905 marked a dramatic turn in our
understanding of space, time, mass, and energy. His questioning of received ideas was most radical.
His construction of alternative theories was most elegant, powerful, and durable. By rejecting the
ether and propounding a new chronogeometry, he prepared the ground for further intellectual
achievements, including general relativity and quantum theory.
Acknowledgments. I thank Thibault Damour and Bertrand Duplantier for useful suggestions.
Short bibliography
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Albert Einstein, “Autobiographisches,” in P.A. Schilpp (éd.), Albert Einstein: Philosopher-scientist
(Evanston, 1949), 1-94.
Jed Buchwald, From Maxwell to microphysics: Aspects of electromagnetic theory in the last quarter
of the nineteenth century (Chicago, 1985).
Olivier Darrigol, Electrodynamics from Ampère to Einstein (Oxford, 2000).
Albrecht Fölsing, Albert Einstein: Eine Biographie (Frankfurt, 1993); transl. (New York, 1993).
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Arthur, Miller, Albert Einstein’s special relativity: Emergence and early interpretation (1905-1911)
(Reading, 1981).
Abraham Pais, “Subtle is the Lord. . . ”: The science and the life of Albert Einstein (Oxford, 1982).
Michel Paty, Einstein philosophe: La physique comme pratique philosophique (Paris, 1993).
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Edmund Whittaker, A history of the theories of aether and electricity, 2 vols. (London, 1951).